Lecture 2 - Solution of First Order Differential Equations | Differential Equation and Mathematical Modeling-II - Engineering Mathematics PDF Download

 Page 1


Solutions of First-Order Differential Equations 
 
 
 
 
 
Discipline Course – I 
Semester - II 
Paper: Differential Equations - I 
Lesson: Solutions of First-Order Differential 
Equations 
Lesson Developer: Anu Sharma 
College:  Aditi Mahavidalaya, University of 
Delhi 
 
 
 
 
 
 
 
 
 
 
 
Institute of Lifelong Learning, University of Delhi                                       pg. 1 
 
Page 2


Solutions of First-Order Differential Equations 
 
 
 
 
 
Discipline Course – I 
Semester - II 
Paper: Differential Equations - I 
Lesson: Solutions of First-Order Differential 
Equations 
Lesson Developer: Anu Sharma 
College:  Aditi Mahavidalaya, University of 
Delhi 
 
 
 
 
 
 
 
 
 
 
 
Institute of Lifelong Learning, University of Delhi                                       pg. 1 
 
Solutions of First-Order Differential Equations 
Table of Contents:  
 Chapter: Solutions of First-Order Differential Equations 
 
• 1. Learning Outcomes  
• 2. Introduction 
• 3. Standard forms of First Order Differential Equations 
o 3.1. Total Differential Function 
o 3.2. Exact Differential Equation 
• Exercise 1 
• 4. Separable Equations 
o 4.1. Solution of Separable Equations 
• Exercises 2 
• 5. Homogeneous Equations 
o 5.1. Solution of Homogeneous Equations 
• Exercises 3 
• 6. Integrating Factors (I.F.) 
• 7. Linear Equations  
o 7.1. Solution of Linear Equation 
o Exercise 4 
o 7.2. Equations reducible to linear Form 
o Exercises 5 
• 8. Special Integrating Factors and Transformations 
o 8.1. Rules for finding Integrating Factors 
• Exercise 6 
• 9.   Special Transformations 
o 9.1. Equations reducible to Homogeneous From 
• Exercise 7 
• Summary 
• References 
 
Institute of Lifelong Learning, University of Delhi                                       pg. 2 
 
Page 3


Solutions of First-Order Differential Equations 
 
 
 
 
 
Discipline Course – I 
Semester - II 
Paper: Differential Equations - I 
Lesson: Solutions of First-Order Differential 
Equations 
Lesson Developer: Anu Sharma 
College:  Aditi Mahavidalaya, University of 
Delhi 
 
 
 
 
 
 
 
 
 
 
 
Institute of Lifelong Learning, University of Delhi                                       pg. 1 
 
Solutions of First-Order Differential Equations 
Table of Contents:  
 Chapter: Solutions of First-Order Differential Equations 
 
• 1. Learning Outcomes  
• 2. Introduction 
• 3. Standard forms of First Order Differential Equations 
o 3.1. Total Differential Function 
o 3.2. Exact Differential Equation 
• Exercise 1 
• 4. Separable Equations 
o 4.1. Solution of Separable Equations 
• Exercises 2 
• 5. Homogeneous Equations 
o 5.1. Solution of Homogeneous Equations 
• Exercises 3 
• 6. Integrating Factors (I.F.) 
• 7. Linear Equations  
o 7.1. Solution of Linear Equation 
o Exercise 4 
o 7.2. Equations reducible to linear Form 
o Exercises 5 
• 8. Special Integrating Factors and Transformations 
o 8.1. Rules for finding Integrating Factors 
• Exercise 6 
• 9.   Special Transformations 
o 9.1. Equations reducible to Homogeneous From 
• Exercise 7 
• Summary 
• References 
 
Institute of Lifelong Learning, University of Delhi                                       pg. 2 
 
Solutions of First-Order Differential Equations 
1. Learning Outcome:  
     After reading this chapter, you should be able to : 
 
(a) Recognize standard forms of First Order Differential Equations 
(b)Exact Differential Equations and their solutions 
(c)Separable Equations and equations reducible to this form and 
their solutions 
(d)Homogeneous Equations and equations reducible to this form 
and their solutions 
(e)Integrating Factors 
(f)Linear Equations and Bernoulli’s Equations and their solutions 
(g)Special Integrating Factors and Transformations 
  
2. Introduction: 
 This chapter deals with first order differential equations for 
which exact solutions can be obtained by definite methods. Here we 
will discuss exact equations, separable equations, homogeneous 
equations, linear equations and Bernoulli's equations. The aim of 
this chapter is to recognize these various types of differential 
equations and solve these by using various methods. Let us discuss 
these equations one by one. 
 
3. Standard forms of First Order Differential Equations 
 The differential equations of first order and first degree are of 
two types namely 
 (, )
dy
f xy
dx
=             (1) 
or the Differential form 
 (, ) (, ) 0 P x y dx Q x y dy +=  (2) 
where P and Q are functions of x and y. 
These two equations clearly state that one of these forms may 
readily be written in the other form. 
For example the equation 
 
1
()
dy x y
dx x y
++
=
-
 is in the differential form 
This equation can be written as 
 ( 1) ( ) 0 x y dx y x dy ++ + - = 
This is the other form of differential equation. 
Similarly, 
Institute of Lifelong Learning, University of Delhi                                       pg. 3 
 
Page 4


Solutions of First-Order Differential Equations 
 
 
 
 
 
Discipline Course – I 
Semester - II 
Paper: Differential Equations - I 
Lesson: Solutions of First-Order Differential 
Equations 
Lesson Developer: Anu Sharma 
College:  Aditi Mahavidalaya, University of 
Delhi 
 
 
 
 
 
 
 
 
 
 
 
Institute of Lifelong Learning, University of Delhi                                       pg. 1 
 
Solutions of First-Order Differential Equations 
Table of Contents:  
 Chapter: Solutions of First-Order Differential Equations 
 
• 1. Learning Outcomes  
• 2. Introduction 
• 3. Standard forms of First Order Differential Equations 
o 3.1. Total Differential Function 
o 3.2. Exact Differential Equation 
• Exercise 1 
• 4. Separable Equations 
o 4.1. Solution of Separable Equations 
• Exercises 2 
• 5. Homogeneous Equations 
o 5.1. Solution of Homogeneous Equations 
• Exercises 3 
• 6. Integrating Factors (I.F.) 
• 7. Linear Equations  
o 7.1. Solution of Linear Equation 
o Exercise 4 
o 7.2. Equations reducible to linear Form 
o Exercises 5 
• 8. Special Integrating Factors and Transformations 
o 8.1. Rules for finding Integrating Factors 
• Exercise 6 
• 9.   Special Transformations 
o 9.1. Equations reducible to Homogeneous From 
• Exercise 7 
• Summary 
• References 
 
Institute of Lifelong Learning, University of Delhi                                       pg. 2 
 
Solutions of First-Order Differential Equations 
1. Learning Outcome:  
     After reading this chapter, you should be able to : 
 
(a) Recognize standard forms of First Order Differential Equations 
(b)Exact Differential Equations and their solutions 
(c)Separable Equations and equations reducible to this form and 
their solutions 
(d)Homogeneous Equations and equations reducible to this form 
and their solutions 
(e)Integrating Factors 
(f)Linear Equations and Bernoulli’s Equations and their solutions 
(g)Special Integrating Factors and Transformations 
  
2. Introduction: 
 This chapter deals with first order differential equations for 
which exact solutions can be obtained by definite methods. Here we 
will discuss exact equations, separable equations, homogeneous 
equations, linear equations and Bernoulli's equations. The aim of 
this chapter is to recognize these various types of differential 
equations and solve these by using various methods. Let us discuss 
these equations one by one. 
 
3. Standard forms of First Order Differential Equations 
 The differential equations of first order and first degree are of 
two types namely 
 (, )
dy
f xy
dx
=             (1) 
or the Differential form 
 (, ) (, ) 0 P x y dx Q x y dy +=  (2) 
where P and Q are functions of x and y. 
These two equations clearly state that one of these forms may 
readily be written in the other form. 
For example the equation 
 
1
()
dy x y
dx x y
++
=
-
 is in the differential form 
This equation can be written as 
 ( 1) ( ) 0 x y dx y x dy ++ + - = 
This is the other form of differential equation. 
Similarly, 
Institute of Lifelong Learning, University of Delhi                                       pg. 3 
 
Solutions of First-Order Differential Equations 
 the equation 
 (sin ) ( 3 ) 0 x y dx x y dy + ++ = 
which is of the form  
 (, ) (, ) 0 P x y dx Q x y dy += 
can be written as 
 
(sin )
( 3 )
dy x y
dx x y
+
= -
+
 
which is the differential form. 
 
3.1. Total Differential Function: 
Let U be a function of x and y, then the total differential dU of the 
function U is defined as 
 
(, ) (, ) U xy U xy
dU dx dy
xy
??
= +
??
 
Example 1: Let U be the function of two variables defined by 
 
23
(, ) 2 U x y x y xy = + 
Then find the total differential of the function U 
Solution: The given function is 
 
23
(, ) 2 U x y x y xy = + 
Then 
 
3
22
U
xy y
x
?
= +
?
 (1) 
and 
22
6
U
x xy
y
?
= +
?
 (2) 
? 
UU
dU dx dy
xy
??
= +
??
  
        
3 22
(2 2 ) ( 6 ) xy y dx x xy dy = + ++ [Using (1) and (2)] 
 
 
 
 
3.2. Exact Differential Equation: 
The differential equation 
 0 Pdx Qdy += 
is said to be exact if 
 Pdx Qdy dU += 
Institute of Lifelong Learning, University of Delhi                                       pg. 4 
 
Page 5


Solutions of First-Order Differential Equations 
 
 
 
 
 
Discipline Course – I 
Semester - II 
Paper: Differential Equations - I 
Lesson: Solutions of First-Order Differential 
Equations 
Lesson Developer: Anu Sharma 
College:  Aditi Mahavidalaya, University of 
Delhi 
 
 
 
 
 
 
 
 
 
 
 
Institute of Lifelong Learning, University of Delhi                                       pg. 1 
 
Solutions of First-Order Differential Equations 
Table of Contents:  
 Chapter: Solutions of First-Order Differential Equations 
 
• 1. Learning Outcomes  
• 2. Introduction 
• 3. Standard forms of First Order Differential Equations 
o 3.1. Total Differential Function 
o 3.2. Exact Differential Equation 
• Exercise 1 
• 4. Separable Equations 
o 4.1. Solution of Separable Equations 
• Exercises 2 
• 5. Homogeneous Equations 
o 5.1. Solution of Homogeneous Equations 
• Exercises 3 
• 6. Integrating Factors (I.F.) 
• 7. Linear Equations  
o 7.1. Solution of Linear Equation 
o Exercise 4 
o 7.2. Equations reducible to linear Form 
o Exercises 5 
• 8. Special Integrating Factors and Transformations 
o 8.1. Rules for finding Integrating Factors 
• Exercise 6 
• 9.   Special Transformations 
o 9.1. Equations reducible to Homogeneous From 
• Exercise 7 
• Summary 
• References 
 
Institute of Lifelong Learning, University of Delhi                                       pg. 2 
 
Solutions of First-Order Differential Equations 
1. Learning Outcome:  
     After reading this chapter, you should be able to : 
 
(a) Recognize standard forms of First Order Differential Equations 
(b)Exact Differential Equations and their solutions 
(c)Separable Equations and equations reducible to this form and 
their solutions 
(d)Homogeneous Equations and equations reducible to this form 
and their solutions 
(e)Integrating Factors 
(f)Linear Equations and Bernoulli’s Equations and their solutions 
(g)Special Integrating Factors and Transformations 
  
2. Introduction: 
 This chapter deals with first order differential equations for 
which exact solutions can be obtained by definite methods. Here we 
will discuss exact equations, separable equations, homogeneous 
equations, linear equations and Bernoulli's equations. The aim of 
this chapter is to recognize these various types of differential 
equations and solve these by using various methods. Let us discuss 
these equations one by one. 
 
3. Standard forms of First Order Differential Equations 
 The differential equations of first order and first degree are of 
two types namely 
 (, )
dy
f xy
dx
=             (1) 
or the Differential form 
 (, ) (, ) 0 P x y dx Q x y dy +=  (2) 
where P and Q are functions of x and y. 
These two equations clearly state that one of these forms may 
readily be written in the other form. 
For example the equation 
 
1
()
dy x y
dx x y
++
=
-
 is in the differential form 
This equation can be written as 
 ( 1) ( ) 0 x y dx y x dy ++ + - = 
This is the other form of differential equation. 
Similarly, 
Institute of Lifelong Learning, University of Delhi                                       pg. 3 
 
Solutions of First-Order Differential Equations 
 the equation 
 (sin ) ( 3 ) 0 x y dx x y dy + ++ = 
which is of the form  
 (, ) (, ) 0 P x y dx Q x y dy += 
can be written as 
 
(sin )
( 3 )
dy x y
dx x y
+
= -
+
 
which is the differential form. 
 
3.1. Total Differential Function: 
Let U be a function of x and y, then the total differential dU of the 
function U is defined as 
 
(, ) (, ) U xy U xy
dU dx dy
xy
??
= +
??
 
Example 1: Let U be the function of two variables defined by 
 
23
(, ) 2 U x y x y xy = + 
Then find the total differential of the function U 
Solution: The given function is 
 
23
(, ) 2 U x y x y xy = + 
Then 
 
3
22
U
xy y
x
?
= +
?
 (1) 
and 
22
6
U
x xy
y
?
= +
?
 (2) 
? 
UU
dU dx dy
xy
??
= +
??
  
        
3 22
(2 2 ) ( 6 ) xy y dx x xy dy = + ++ [Using (1) and (2)] 
 
 
 
 
3.2. Exact Differential Equation: 
The differential equation 
 0 Pdx Qdy += 
is said to be exact if 
 Pdx Qdy dU += 
Institute of Lifelong Learning, University of Delhi                                       pg. 4 
 
Solutions of First-Order Differential Equations 
or 
 
UU
Pdx Qdy dx dy
xy
??
+= +
??
 [by the def. of total differential] 
or 
  and 
UU
pQ
xy
??
= =
??
 
Theorem 1: Find the necessary and sufficient condition for the 
equation Pdx Qdy += 0 to be exact. 
Proof. (i)  To find the necessary condition 
 Let Pdx Qdy += 0 be exact. (1) 
Then by the definition of exact differential equation 
 , Pdx Qdy dU += where U is function of x and y. (2) 
or 
 
UU
Pdx Qdy dx dy
xy
??
+= +
??
 (3) 
Equating coefficients of dx and dy in (3), we get 
 
U
P
x
?
=
?
  (4) 
and  
 
U
Q
y
?
=
?
  (5) 
Differentiate (4) and (5) partially w.r.t. y and x respectively. 
 
2
U
==
yx
P U
y yx
? ? ? ?
??
??
? ? ? ??
??
 (6) 
and 
 
2
U
==
xy
QU
x xy
?? ? ?? ?
??
? ? ? ??
??
 (7) 
But 
 
22
UU
y x xy
??
=
? ? ??
 
From (6) and (7), we get 
 =
PQ
yx
??
??
 
which is the required necessary condition. 
(ii)  To prove that the condition is sufficient 
Institute of Lifelong Learning, University of Delhi                                       pg. 5